Needing a long zoom

I shoot a Nikon d5200 and a Nikon d800. The first is a crop sensor and the latter full frame. I'm leaning toward buying a f4 70-200 for about $1400.00 plus tax. Money wise I was considering the f4.5-5.6 70-300 for a lot less $ and using it only on my d5200. But with the f4 70-200 I can have better results on both camera bodies. I need it for outdoors, nature, animals, birds, kids and people in general and actual posed portraits. My dilemma is - what do you think between the f2.8 and the f4 70-200's? For the $1000.00 savings will I really miss or need the larger aperture of the 2.8 and it's DOF? I like the idea of having the 300mm focal length on my d5200 body. I will be printing 11x14 or larger from this lens, and from both bodies.Chip